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Up to this point the square root of a negative number has been left undefined. For example, we know that is not a real number.
There is no real number that when squared results in a negative number. We begin to resolve this issue by defining the imaginary unitDefined as where , i, as the square root of −1.
To express a square root of a negative number in terms of the imaginary unit i, we use the following property where a represents any non-negative real number:
With this we can write
If , then we would expect that squared will equal −9:
In this way any square root of a negative real number can be written in terms of the imaginary unit. Such a number is often called an imaginary numberA square root of any negative real number..
Rewrite in terms of the imaginary unit i.
Solution:
Notation Note: When an imaginary number involves a radical, we place i in front of the radical. Consider the following:
Since multiplication is commutative, these numbers are equivalent. However, in the form , the imaginary unit i is often misinterpreted to be part of the radicand. To avoid this confusion, it is a best practice to place i in front of the radical and use
A complex numberA number of the form where a and b are real numbers. is any number of the form, where a and b are real numbers. Here, a is called the real partThe real number a of a complex number and b is called the imaginary partThe real number b of a complex number . For example, is a complex number with a real part of 3 and an imaginary part of −4. It is important to note that any real number is also a complex number. For example, 5 is a real number; it can be written as with a real part of 5 and an imaginary part of 0. Hence, the set of real numbers, denoted , is a subset of the set of complex numbers, denoted
Complex numbers are used in many fields including electronics, engineering, physics, and mathematics. In this textbook we will use them to better understand solutions to equations such as For this reason, we next explore algebraic operations with them.
Adding or subtracting complex numbers is similar to adding and subtracting polynomials with like terms. We add or subtract the real parts and then the imaginary parts.
Add:
Solution:
Add the real parts and then add the imaginary parts.
Answer:
To subtract complex numbers, we subtract the real parts and subtract the imaginary parts. This is consistent with the use of the distributive property.
Subtract:
Solution:
Distribute the negative sign and then combine like terms.
Answer:
In general, given real numbers a, b, c and d:
Simplify:
Solution:
Answer:
In summary, adding and subtracting complex numbers results in a complex number.
Multiplying complex numbers is similar to multiplying polynomials. The distributive property applies. In addition, we make use of the fact that to simplify the result into standard form
Multiply:
Solution:
We begin by applying the distributive property.
Answer:
Multiply:
Solution:
Answer:
In general, given real numbers a, b, c and d:
Given a complex number , its complex conjugateTwo complex numbers whose real parts are the same and imaginary parts are opposite. If given , then its complex conjugate is is We next explore the product of complex conjugates.
Multiply:
Solution:
Answer: 29
In general, the product of complex conjugatesThe real number that results from multiplying complex conjugates: follows:
Note that the result does not involve the imaginary unit; hence, it is real. This leads us to the very useful property
To divide complex numbers, we apply the technique used to rationalize the denominator. Multiply the numerator and denominator by the conjugate of the denominator. The result can then be simplified into standard form
Divide:
Solution:
In this example, the conjugate of the denominator is Therefore, we will multiply by 1 in the form
To write this complex number in standard form, we make use of the fact that 13 is a common denominator.
Answer:
Divide:
Solution:
Answer:
In general, given real numbers a, b, c and d where c and d are not both 0:
Divide:
Solution:
Here we can think of and thus we can see that its conjugate is
Because the denominator is a monomial, we could multiply numerator and denominator by 1 in the form of and save some steps reducing in the end.
Answer:
When multiplying and dividing complex numbers we must take care to understand that the product and quotient rules for radicals require that both a and b are positive. In other words, if and are both real numbers then we have the following rules.
For example, we can demonstrate that the product rule is true when a and b are both positive as follows:
However, when a and b are both negative the property is not true.
Here and both are not real numbers and the product rule for radicals fails to produce a true statement. Therefore, to avoid some common errors associated with this technicality, ensure that any complex number is written in terms of the imaginary unit i before performing any operations.
Multiply:
Solution:
Begin by writing the radicals in terms of the imaginary unit i.
Now the radicands are both positive and the product rule for radicals applies.
Answer:
Multiply:
Solution:
Begin by writing the radicals in terms of the imaginary unit and then distribute.
Answer:
In summary, multiplying and dividing complex numbers results in a complex number.
Rewrite in terms of imaginary unit i.
Write the complex number in standard form
Given that compute the following powers of
Perform the operations.
Perform the operations.
Given that compute the following powers of
Perform the operations and simplify.
Show that both and satisfy
Show that both and satisfy
Show that both and satisfy
Show that both and satisfy
Show that 3, , and are all solutions to
Show that −2, , and are all solutions to
Research and discuss the history of the imaginary unit and complex numbers.
How would you define and why?
Research what it means to calculate the absolute value of a complex number Illustrate your finding with an example.
Explore the powers of i. Look for a pattern and share your findings.
2
2
20
22
0
−3
Proof
Proof
Proof
Answer may vary
Answer may vary